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The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers to gather their reasonings and understandings of each problem.

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PREFACE:

PREFACE

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the productions of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth — the very experience out of which Love rises. - Plotinus, The Enneads, 11.9.16 A: Plotinus

This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere. The most important change is that I have included material on hyperbolic geometry that was missing in the first book. This has also necessitated more discussions of circles and their properties. In addition, there is added material on geometric manifolds and the shape of space. I decided to include hyperbolic geometry for two reasons: 1) the cosmologists say that our physical universe very likely has (at least in part) hyperbolic geometry, and 2) Daina Taimina, a mathematician at the University of Latvia and now my wife, figured out how to crochet a hyperbolic plane, which allowed us to explore intuitively for the first time the geometry of the hyperbolic plane. In addition, Daina Taimina has been responsible for including in this edition significantly more historical material. In this historical material we discuss and try to clear up many current misconceptions that are commonly heldabout some mathematical ideas.

This book is based on a junior/senior-level course I have been teaching since 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication. Certain problems and sections in this book require from the reader a background more advanced than first-semester calculus. These sections are indicated with an asterisk (*) and the background required is indicated (usually at the beginning of the chapter).

The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems, and are encouraged to write and speak their seasonings and understandings. I listen to and critique their thinking and use it to stimulate whole class discussions.

The formal expression of "straightness" is a very difficult formal area of mathematics. However, the concept of "straight" an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the (Euclidean) plane, on a sphere, or on a hyperbolic plane. We will study these ideas and questions, as much as is possible, from an intrinsic point-of-view — that is, the point-of-view of a 2-dimensional bug crawling around on the surface. This will lead to the question: "What is the shape of our physical three-dimensional universe?" Here we are like 3-dimensional bugs who can only view the universe intrinsically.

Most of the problems are approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). I find that by exploring the geometry of a sphere and a hyperbolic plane my students gain a deeper understanding of the geometry of the (Euclidean) plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. I find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. (All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.)

USEFUL SUPPLEMENTS

A faculty member may obtain from the publisher the Instructor's Manual (containing possible solutions to each problem and discussions on how to use this book in a course) by sending a request via e-mail to George_Lobell@prenhall.com or calling 1-201-236-7407.

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase Lénárt Sphere® sets — a transparent sphere, a spherical compass, and a spherical "straight edge" that doubles as a protractor. They work well for small group explorations in the classroom and are available from Key Curriculum Press. However, considerably less expensive alternatives are available: A beach ball or basketball will work for classroom demonstrations, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles. Also, many craft stores carry inexpensive plastic spheres that can be used successfully.

I strongly urge that you have a hyperbolic surface such as those described in Chapter 5. Unfortunately, such hyperbolic surfaces are not readily available commercially. However, directions for making such surfaces (out of paper or by crocheting) are contained in Chapter 5, and I will list patterns for making paper models and sources for crocheted hyperbolic surfaces at:

www.math.cornell.edu/~dwh/books/eg00/supplements.html

as they become available. Most books that explore hyperbolic geometry do so by considering only one of the various "models" of hyperbolic geometry, which give representations of hyperbolic geometry in the same way that a map of a portion of the earth gives a representation of a portion of the earth. Each of these representations necessarily (see Chapter 16) distorts either straight lines or angles or both.

In addition, the use of dynamic geometry software such as Geometers Sketchpad®,Cabri®, or Cinderella® will enhance any geometry course. These software packages were originally written for exploring Euclidean plane geometry, but recent versions allow one to also dynamically explore spherical and hyperbolic geometries. I will maintain at the web address listed above links to information about these software packages and to web pages that give examples on how to use them for self-learning or in a classroom.

MY TEACHING BACKGROUND

My teaching is a product of Western Civilization. My known ancestors lived in England, Scotland, Ireland, Germany, and Luxembourg and I am a descendent from a long line of academics stretching back (according to family traditions) to at least the seventeenth century. My mode of teaching also has deep Western roots that reach back to the Socratic dialogues recorded by Plato in ancient Greece. More directly, my teaching has been influenced by my experiences in high school, college, and graduate school. In Ames, Iowa, my high school world literature teacher, Mary McNally, coaxed deep creative thinking out of us through her many writing assignments which she read with great interest in our ideas. At Swarthmore College in Pennsylvania instead of classes I spent my last two years in student participation seminars and tutorials, where I learned to take charge of my own learning and become an academic scholar. In graduate school at the University of Wisconsin my mentor, R.H. Bing, taught without lectures or textbooks in a style which is often known as the Moore Method, named after Bing's graduate mentor R.L. Moore at the University of Texas. (See TG: Traylor for more information on the Moore Method.) R.L. Moore received his PhD at the University of Chicago before the turn of the century and was one of the very first Americans to receive a PhD in mathematics in this country. My teaching of the geometry course and the writing of this book evolved from this background.

A Short History of Hyperbolic Geometry. Constructions of Hyperbolic Planes. Hyperbolic Planes of Different Raddi (Curvature). Problem 5.1: What Is Straight in a Hyperbolic Plane? Problem 5.2: The Pseudosphere Is Hyperbolic. Problem 5.3: Rotations and Reflections on Surfaces.

Problem 7.1: The Area of a Triangle on a Sphere. Problem 7.2: Area of Hyperbolic Triangles. Problem 7.3: Sum of the Angles of a Triangle. Introduction Parallel Transport and Holonomy. Problem 7.4: The Holonomy of a Small Triangle. The Gauss-Bonnet Formula for Triangles. Problem 7.5: Gauss-Bonnet Formula for Polygons. Gauss-Bonnet Formula for Polygons on Surfaces.

Square Roots. Problem 13.1: A Rectangle Dissects into a Square. Baudhayana's Sulbasutram. Problem 13.2: Equivalence of Squares. Any Polygon Can Be Dissected into a Square. Problem 13.3: Similar Triangles. Three-Dimensional Dissections and Hilbert's Third Problem.

Problem 18.1: Quadratic Equations. Problem 18.2: Conic Sections and Cube Roots. Problem 18.3: Roots of Cubic Equations. Problem 18.4: Algebraic Solution of Cubics. So What Does This All Point To?

19. Trigonometry and Duality.

Problem 19.1: Circumference of a Circle. Problem 19.2: Law of Cosines. Problem 19.3: Law of Sines. Duality on a Sphere. Problem 19.4: The Dual of a Small Triangle. Problem 19.5: Trigonometry with Congruences. Duality on the Projective Plane. Problem 19.6: Properties on the Projective Plane. Perspective Drawings and Vision.

Preface

PREFACE:

PREFACE

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the productions of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth — the very experience out of which Love rises. - Plotinus, The Enneads, 11.9.16 A: Plotinus

This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere. The most important change is that I have included material on hyperbolic geometry that was missing in the first book. This has also necessitated more discussions of circles and their properties. In addition, there is added material on geometric manifolds and the shape of space. I decided to include hyperbolic geometry for two reasons: 1) the cosmologists say that our physical universe very likely has (at least in part) hyperbolic geometry, and 2) Daina Taimina, a mathematician at the University of Latvia and now my wife, figured out how to crochet a hyperbolic plane, which allowed us to explore intuitively for the first time the geometry of the hyperbolic plane. In addition, Daina Taimina has been responsible for including in this edition significantly more historical material. In this historical material we discuss and try to clear up many current misconceptions that are commonlyheldabout some mathematical ideas.

This book is based on a junior/senior-level course I have been teaching since 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication. Certain problems and sections in this book require from the reader a background more advanced than first-semester calculus. These sections are indicated with an asterisk (*) and the background required is indicated (usually at the beginning of the chapter).

The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems, and are encouraged to write and speak their seasonings and understandings. I listen to and critique their thinking and use it to stimulate whole class discussions.

The formal expression of "straightness" is a very difficult formal area of mathematics. However, the concept of "straight" an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the (Euclidean) plane, on a sphere, or on a hyperbolic plane. We will study these ideas and questions, as much as is possible, from an intrinsic point-of-view — that is, the point-of-view of a 2-dimensional bug crawling around on the surface. This will lead to the question: "What is the shape of our physical three-dimensional universe?" Here we are like 3-dimensional bugs who can only view the universe intrinsically.

Most of the problems are approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). I find that by exploring the geometry of a sphere and a hyperbolic plane my students gain a deeper understanding of the geometry of the (Euclidean) plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. I find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. (All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.)

USEFUL SUPPLEMENTS

A faculty member may obtain from the publisher the Instructor's Manual (containing possible solutions to each problem and discussions on how to use this book in a course) by sending a request via e-mail to George_Lobell@prenhall.com or calling 1-201-236-7407.

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase Lénárt Sphere® sets — a transparent sphere, a spherical compass, and a spherical "straight edge" that doubles as a protractor. They work well for small group explorations in the classroom and are available from Key Curriculum Press. However, considerably less expensive alternatives are available: A beach ball or basketball will work for classroom demonstrations, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles. Also, many craft stores carry inexpensive plastic spheres that can be used successfully.

I strongly urge that you have a hyperbolic surface such as those described in Chapter 5. Unfortunately, such hyperbolic surfaces are not readily available commercially. However, directions for making such surfaces (out of paper or by crocheting) are contained in Chapter 5, and I will list patterns for making paper models and sources for crocheted hyperbolic surfaces at:

www.math.cornell.edu/~dwh/books/eg00/supplements.html

as they become available. Most books that explore hyperbolic geometry do so by considering only one of the various "models" of hyperbolic geometry, which give representations of hyperbolic geometry in the same way that a map of a portion of the earth gives a representation of a portion of the earth. Each of these representations necessarily (see Chapter 16) distorts either straight lines or angles or both.

In addition, the use of dynamic geometry software such as Geometers Sketchpad®,Cabri®, or Cinderella® will enhance any geometry course. These software packages were originally written for exploring Euclidean plane geometry, but recent versions allow one to also dynamically explore spherical and hyperbolic geometries. I will maintain at the web address listed above links to information about these software packages and to web pages that give examples on how to use them for self-learning or in a classroom.

MY TEACHING BACKGROUND

My teaching is a product of Western Civilization. My known ancestors lived in England, Scotland, Ireland, Germany, and Luxembourg and I am a descendent from a long line of academics stretching back (according to family traditions) to at least the seventeenth century. My mode of teaching also has deep Western roots that reach back to the Socratic dialogues recorded by Plato in ancient Greece. More directly, my teaching has been influenced by my experiences in high school, college, and graduate school. In Ames, Iowa, my high school world literature teacher, Mary McNally, coaxed deep creative thinking out of us through her many writing assignments which she read with great interest in our ideas. At Swarthmore College in Pennsylvania instead of classes I spent my last two years in student participation seminars and tutorials, where I learned to take charge of my own learning and become an academic scholar. In graduate school at the University of Wisconsin my mentor, R.H. Bing, taught without lectures or textbooks in a style which is often known as the Moore Method, named after Bing's graduate mentor R.L. Moore at the University of Texas. (See TG: Traylor for more information on the Moore Method.) R.L. Moore received his PhD at the University of Chicago before the turn of the century and was one of the very first Americans to receive a PhD in mathematics in this country. My teaching of the geometry course and the writing of this book evolved from this background.

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